Calculus: Volumes of Cross Sections
Cross Sections
Find the volume of the solid generated by each cross section.
1. Let R be the region bounded by y = sinx and y = cosx for 0  x 

. The solid has a base in the region R
4
whose cross sections are cut by planes perpendicular to the x-axis which is a square.
1
2. Let R be the region bounded by y 
for 4  x  9 in the 1st Quadrant. The solid has a base in the
x
region R whose cross sections are cut by planes perpendicular to the x-axis which is a rectangle. The
rectangles have heights that are twice their base.
3. Let R be the region bounded by y  x  3 , y = 0, and x = 6. The solid has a base in the region R whose
cross sections are cut by planes perpendicular to the x-axis which is a semicircle.
x
4. The base of a solid is the 1st bounded region R in the in the 1st quadrant, bounded by y  e and
2
y = 1 – cosx . The cross section of R perpendicular to the x-axis is a right triangle whose height is three
times its base. The base of each triangle is in the xy-plane.
5. The base of a solid is the region R bounded by y  ln( x 2  1) and y = cosx. Each cross section of the
solid perpendicular to the x-axis is an equilateral triangle.
6. The base of a solid S is enclosed by y = 1-sinx , x-axis, and y-axis. The cross section of S perpendicular
to the x-axis is an isosceles right triangle whose hypotenuse lies in the xy-plane.
7. Let S be one of the regions bounded by the x-axis and y  cos x . The region S is the base of a solid
whose cross section perpendicular to the x-axis is a right triangle with base and height equal. The base
of each triangle is in the xy-plane.